We first consider a case with a point robot and no uncertainty in obstacle location. The planned trajectory in this case is seen in cyan in Fig. REF (a) with total planning time of \(0.0051s(\pm 0.0008s)\) . Please note that the total planning time also includes the collision probability computation time. Next, we consider uncertainty in one of the obstacle location, whose covariance ellipse is shown in gray. The planned trajectory is seen in cyan in Fig. REF (b) and the planning was completed under \(0.0279s(\pm 0.0043s)\) . Due to the uncertainty in the obstacle location, the robot takes a longer route to avoid collision. A robot of radius 0.3\(m\)  and certain (negligible uncertainty) obstacles gave the same trajectory as in Fig. REF (a) with a planning time of \(0.0055s(\pm 0.0009s)\) . However, when the obstacle location is uncertain the resulting trajectory is as shown in Fig. REF (c). A change in the trajectory is observed, as compared to the case of a point robot in Fig. REF (b). The planning time in this case is \(0.0294s(\pm 0.0047s)\) . It is also worth mentioning that in Fig. REF (b) and (c), the roadmap was updated by adding a node since a \(0.99-\) safe configuration could not be found. The added node is seen in brown, with its coordinates being approximately \((9,11)\) . We also run the case with no obstacle uncertainty and a robot of radius 0.3\(m\)  using the approach of Park et al. [1]}. In this case the planned trajectory is as given in Fig. REF (d). Note that using our approach, the same scenario gives a shorter trajectory (Fig. REF (a)). The longer trajectory computed using the approach in [1]} is due to the fact that a loose upper bound is computed for the collision probability. As a result a longer trajectory is obtained. Contrary to this, we compute the exact collision probability and hence a shorter trajectory is synthesized. The same scenario is also run with the approach in [3]} and produced a trajectory similar to ours. However, since the uncertainties are significantly lower, the approximate collision probability values computed using [3]} are much smaller than the actual values.
